Some Diophantine equations involving arithmetic functions and Bhargava factorials

F. Luca proved for any fixed rational number >0 that the Diophantine equations of the form \, m!=f (n!), where f is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions (m, n). In this paper we generalize the mentioned result and show that Diophantine equations of the form \, m₁! mᵣ!=f (n!) have finitely many integer solutions, too. In addition, we do so by including the case f is the sum of kth powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.